The Turkish club, who finished fourth in the Süper Lig last season, have already had an opening offer of around €12m-€13m knocked back, with Newcastle United demanding closer to €23m for the 26-year-old, who is entering the final year of his contract at St James' Park.
Willock joined Newcastle from Arsenal for £25m in 2021, and while his preference has reportedly been to stay and fight for his place, the growing competition in Newcastle's midfield means a departure remains a live possibility before the window closes, with Beşiktaş still hopeful of finding a compromise closer to €15m.
That competition for places is set to increase further, with Newcastle continuing to work on a £13m deal to sign Højbjerg from Marseille.
The Denmark international would provide the kind of experienced leader new mamager Matthias Jaissle wants alongside summer arrivals Sean Steur and Aladji Bamba, both of whom have joined the club's midfield ranks this window following the departures of Sandro Tonali and Bruno Guimarães.
With Joelinton and Lewis Miley also part of the picture, Newcastle's midfield is increasingly crowded, adding further weight to the sense that Willock's future may lie away from Tyneside this summer.